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Operators on Tensor States

  1. Oct 10, 2009 #1

    My brain seems to have shut down. Lets say I'm working in the space [itex] H_a \otimes H_b [/itex] and I have an operator A and B on each of these spaces respectively. Furthermore, consider a general state in the tensor space [itex] |\psi \rangle = | x_1 \rangle |y_1\rangle + |x_2 \rangle |y_2 \rangle [/itex] (ignoring normalization). Now in my head, I keep on thinking that if I wanted to apply the operator [itex] A \otimes B [/itex] to [itex] | \psi \rangle [/itex] I couldn't simply go

    [tex] A \otimes B | \psi \rangle = A| x_1 \rangle B|y_1\rangle + A|x_2 \rangle B|y_2 \rangle [/tex]

    However, I was playing around with some values today (assuming finite dimensions), and from what I tried it always seemed to work. Can we do this in general, or did I just get lucky in all my trials?
  2. jcsd
  3. Oct 10, 2009 #2
    It's fine. First, AxB is defined so that (AxB)(|y> x |z>)=(A|y> x B|z>), and second, the operator (AxB) is linear on the tensor space...call [tex](A\times B)=O[/tex] and [tex]|y> \times |z>=\Psi[/tex]. Then [tex]O (\Psi_1+\Psi_2)=(O\Psi_{1}+O\Psi_{2})[/tex]. Then use the first property for each of the terms. (the "x" is product)
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